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7f83440352d1bd04342f5f3ef5ea9877b179c5c2 | subsection | 64 | 279 | Sketch of isomorphism between the two Floer complexes | Observe that they are equal if H is z-independent and \beta = dt.Lemma 5.4
Let (W, d\lambda ) be an exact symplectic manifold with cylindrical end ((r_0, +\infty ) \times Y, d( \operatorname{e}^r \alpha )), r_0 < \log 2.Let (\beta ,H, J) be a monotone triple on the cylinder (\mathbb {R}\times S^1, i) with 1-form \beta... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
a0318b97bf7bfc701b098b255477f99632a1e2cd | subsection | 65 | 279 | Sketch of isomorphism between the two Floer complexes | Take a regular value r_1-c of \tilde{r} \circ \tilde{v} \colon \mathbb {R}\times S^1 \rightarrow \mathbb {R} in the interval (r_1-\delta /2, r_1).By hypothesis, there exists a k_0 sufficiently large so that for each k \ge k_0, \min _{t \in S^1} \tilde{r}( \tilde{v}(s_{k}, t) ) > r_1-c/2. Fix such a k.Define the subdoma... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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ee3b935be32d746961800afb56f9764fdcc77ac2 | subsection | 66 | 279 | Sketch of isomorphism between the two Floer complexes | Combining Stokes's Theorem with Lemma REF (and implicitly using biholomorphisms between neighborhoods of the connected components of \Gamma in S_k and annuli of the form (-\epsilon ,0]\times S^1), we obtainE_{\mathrm {topo}} ( \tilde{v}|_{S^k} ) &= \int _{ \lbrace s_k \rbrace \times S^1 } \tilde{v}^*\lambda - H(s,t,\ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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433ed2315eb999ec1a850cd9b16f7f9f48b309e4 | subsection | 67 | 279 | Sketch of isomorphism between the two Floer complexes | As S_k has non-empty interior, we conclude that \partial _s\tilde{v} = 0, which contradicts the fact that \tilde{v}(s,t) \rightarrow \gamma _- as s \rightarrow -\infty .We now explain how the previous result completes the argument that none of the problematic configurations described earlier occur, and thus \Phi is a c... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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cdc2e707bcb7fcb1bc6efd18576017bcb949762a | subsection | 68 | 279 | A correspondence between Floer and pseudoholomorphic cylinders | The content of this section is technical, so we begin
with an overview and explanation of its relevance to the study of the split symplectic
homology differential. The main results are Propositions REF
and REF , which we summarize in the following result. For precise
statements, notably of the uniqueness, see the full... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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c9dcd74e7709a3f1e4679458e5f66019a249a410 | subsection | 69 | 279 | A correspondence between Floer and pseudoholomorphic cylinders | Recall that \sbox
{ \mathcal {M}}\widetilde{\usebox {
}}^*_{H,0,\mathbb {R}\times Y;k_-,k_+}(A;J_Y) is the
space of parametrized unpunctured Floer cylinders \tilde{v}, going from orbits of multiplicity k_- (as s\rightarrow -\infty ) to orbits
of multiplicity k_+ (as s\rightarrow +\infty ), where k_+>k_-.
The fibre pro... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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88674b784d9aba0e982e3bb5081cfe31d7ac5432 | subsection | 70 | 279 | A correspondence between Floer and pseudoholomorphic cylinders | Hence, U is asymptotic to a Reeb orbit of multiplicity
k_+-k_- = B\bullet \Sigma over p. Proposition REF now implies that \sbox
{ \mathcal {M}}\widetilde{\usebox {
}}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y)
can be identified with the space of punctured J_Y-holomorphic cylinders \mathbb {R}\times S^1 \setminus \lbra... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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b3e9ba03372a7034f778f132ef915dab7182dbef | subsection | 71 | 279 | A correspondence between Floer and pseudoholomorphic cylinders | The main results are Propositions REF
and REF , which we summarize in the following result. For precise
statements, notably of the uniqueness, see the full statements of the
propositions in the later sections.Proposition 6.1
There is a bijection\lbrace \tilde{v}\colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.... |
3e7719c0ad0fb36b04f6d66809d435efb4f11c0a | subsection | 72 | 279 | A correspondence between Floer and pseudoholomorphic cylinders | Recall that \sbox
{ \mathcal {M}}\widetilde{\usebox {
}}^*_{H,0,\mathbb {R}\times Y;k_-,k_+}(A;J_Y) is the
space of parametrized unpunctured Floer cylinders \tilde{v}, going from orbits of multiplicity k_- (as s\rightarrow -\infty ) to orbits
of multiplicity k_+ (as s\rightarrow +\infty ), where k_+>k_-.
The fibre pro... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.005... |
e2d16cfe10d4d78aca9cf2eb79d7d6f5fb85e1e5 | subsection | 73 | 279 | A correspondence between Floer and pseudoholomorphic cylinders | Hence, U is asymptotic to a Reeb orbit of multiplicity
k_+-k_- = B\bullet \Sigma over p. Proposition REF now implies that \sbox
{ \mathcal {M}}\widetilde{\usebox {
}}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y)
can be identified with the space of punctured J_Y-holomorphic cylinders \mathbb {R}\times S^1 \setminus \lbra... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
e1d7e303c13f2b6254d58f72b95668e4e2cd85ac | subsection | 74 | 279 | Sobolev spaces and Morse–Bott Riemann–Roch | Before we state and prove the main results in this section, we need to briefly set-up
the appropriate Fredholm theory. We refer to
*Section 5.2 for more details.We will need to consider exponentially weighted Sobolov spaces of sections of the trivial complex
line bundle over a Riemann surface \dot{S}= \mathbb {R}\times... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.044545140117406845,
0.013470328412950039,
0.0... |
928887d6f657b216f4d81b3f0bee84312f94ab12 | subsection | 75 | 279 | Sobolev spaces and Morse–Bott Riemann–Roch | This path has a
Conley–Zehnder index, that we denote by CZ(\mathbf {A}_z + \delta )
(see also *Section 5.2.1).Theorem 6.2
Let \delta > 0 be sufficiently small that
for each puncture z \in \Gamma \cup \lbrace \pm \infty \rbrace ,
[-\delta , \delta ]\cap \sigma (\mathbf {A}_z) \subset \lbrace 0 \rbrace .
Let \mathbf {V}... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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2366f4895bcf2adc93c7a0d49e0ae3f333e834ef | subsection | 76 | 279 | Sobolev spaces and Morse–Bott Riemann–Roch | If C>0, then the same is true except for the eigenvalues -C and 0, corresponding to k=0 above, both of which have multiplicity 1.In particular, the \sigma (A_0) = 2\pi \mathbb {Z} and the winding number of 2\pi k is k.Corollary 6.4
Take C\ge 0 and \delta >0 such that [-\delta ,\delta ] \cap \sigma (A_C) = \lbrace 0\rb... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0... |
dcfc9f41e23bf6dd0b8bd14fda5bec6c5f430d74 | subsection | 77 | 279 | Sobolev spaces and Morse–Bott Riemann–Roch | For \delta >0, we then consider the space W^{1,p,\delta }_{\mathbf {V}}(\dot{S},
of sections that converge exponentially at each puncture z to a vector
in the corresponding vector space V_z.Denote the spectrum of an asymptotic operator \mathbf {A}_z as
above by \sigma (\mathbf {A}_z). The operator \mathbf {A}_z is of
... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-0.003... |
19757987f18c2d242d0d402a3109b001290a8bd8 | subsection | 78 | 279 | Sobolev spaces and Morse–Bott Riemann–Roch | Let \mathbf {V} be a collection of vector spaces, associating to each
z \in \Gamma \cup \lbrace \pm \infty \rbrace the vector subspace V_z \subset \ker \mathbf {A}_z.Then,D \colon W^{1,p, {\delta }}_{\mathbf {V}}(\dot{S}, \rightarrow L^{p, \mathbf {\delta }}(\dot{S}, \Lambda ^{0,1} T^*\dot{S}).is Fredholm, and its Fred... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
4b143f9d60ecd0914606294da3fd0e14af650aa6 | subsection | 79 | 279 | From Floer cylinders to pseudoholomorphic curves | We show that every Floer cylinder as in (REF )
can be obtained from a J_Y-holomorphic curve and a solution
to an auxiliary equation.It is useful to first recall that for every integer T \ge 1,
there is a Y-family of Reeb orbits of period T, and a bijective
correspondence to the Y-family of 1-periodic Hamiltonian orbits... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
5207464208603849801cf5673214263b3fb42d57 | subsection | 80 | 279 | From Floer cylinders to pseudoholomorphic curves | The original Floer solution is given by
(b, v) = (a + e_1, \phi _R^{e_2} \circ u )
where \phi _R^t \colon Y\rightarrow Y denotes Reeb flow for time t\in S^1.The pair (\tilde{u},{\bf e}) is unique, up to replacing \tilde{u} = (a,u) with (a+c_1,u) and replacing {\bf e}= (e_1,e_2) with (e_1 - c_1, e_2), for some constan... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0... |
7f4324e8287bd05c9d98c25b331ea28ba4df2c0d | subsection | 81 | 279 | From Floer cylinders to pseudoholomorphic curves | Recall that the space of functions W^{1,p,\delta }_{(0,}( \mathbb {R}\times S^1, consists of complex-valued functions
exponentially decaying to 0 at -\infty and exponentially decaying
to an unspecified constant at +\infty . In the case of a punctured
cylinder, W^{1,p,\delta }_{(0, }(\mathbb {R}\times S^1 \setminus \lbr... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
22e6ad52aeabe39834ec69b98d3ce61cb6326222 | subsection | 82 | 279 | From Floer cylinders to pseudoholomorphic curves | We say \tilde{\mathbf {e}}\in \tilde{\mathcal {X}}^1 if
\tilde{\mathbf {e}}+ (T_+ s + i 0) \, \mu _{+\infty } + (T_- s + i 0) \, \mu _{-\infty } \in W^{1,p, \delta }_{( } (\mathbb {R}\times S^1, .
Case 2:
\Gamma = \lbrace P\rbrace and both asymptotic limits of \tilde{v} are Hamiltonian orbits.
We say \tilde{\mathbf {... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
db22771fc3adfd339e15cd6be1b25ca1b7c5d51f | subsection | 83 | 279 | From Floer cylinders to pseudoholomorphic curves | We say \tilde{\mathbf {e}}\in \tilde{\mathcal {X}}^3 if
\tilde{\mathbf {e}}+ (T_+ s + i 0) \, \mu _{+\infty } \in W^{1,p\delta }_{( }( \mathbb {R}\times S^1, .We then define \mathcal {X}^1, \mathcal {X}^2, \mathcal {X}^3 to be the set of functions obtained as compositions of functions in \tilde{\mathcal {X}}^1, \tilde... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.002... |
82499d3ffb8cb7f0edcae587ed498e74b4ac3db3 | subsection | 84 | 279 | From Floer cylinders to pseudoholomorphic curves | The corresponding linearized operator is\begin{aligned}D^j \colon T_{\bf e} \mathcal {X}^j &\rightarrow L^{p,\delta }(T^*( \mathbb {R}\times S^1 \setminus \Gamma ) ) \\
E = E_1 + i E_2 &\mapsto dE_1 - d E_2 \circ i
\end{aligned}where
in Cases 1 and 3, we have T_{\bf e} \mathcal {X}^j = W^{1,p,\delta }_{( }(\mathbb {R}\... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.021208... |
ed048a5cd243afc5991b44bdca95b40797cfd6dd | subsection | 85 | 279 | From Floer cylinders to pseudoholomorphic curves | Its asymptotic operator is again A. The Euler characteristic of the domain is now -1.
Therefore,\operatorname{Ind}D^{2} = -1 + \big ((-1+2)-(-1)-(-1)\big ) = 2,c_1(E,l,\textbf {A}_\Gamma ) = \frac{1}{2}(2-2) = 0 < 2 = \operatorname{Ind}\widehat{D}^{2}and we can once more apply *Proposition 2.2 together with
*Lemma 5.20... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.00578259... |
4e882bf5708ac48dd9f72b0c490e817ca1b0e473 | subsection | 86 | 279 | From Floer cylinders to pseudoholomorphic curves | Then,\int _{\partial S_c} \tilde{e}_2 \, dt
&= \int _{S_c} d\tilde{e}_2 \wedge dt = \int _{S_c} de_2 \wedge dt \\
&= \int _{S_c} de_2 \circ i \wedge dt\circ i \\
&= \int _{S_c} ( de_1 + h^{\prime }(\operatorname{e}^{b})ds ) \wedge ds \\
&= \int _{S_c} de_1 \wedge ds \\
&= \int _{\partial S_c} e_1 \, ds =0since ds|_{\pa... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-0.... |
883be3c1f8ab8f19f7373a79d41ce39bd51b2473 | subsection | 87 | 279 | From Floer cylinders to pseudoholomorphic curves | Let \nu _j: \mathbb {R}\rightarrow \mathbb {R} be smooth functions such that{\left\lbrace \begin{array}{ll}
\nu _j(s) = T_- s \text{ for } s<<0 \text{ and } \nu _j(s) = T_+ s \text{ for } s>>0 & j = 1, 2 \\
\nu _j(s) = 0 \text{ for } s<<0 \text{ and } \nu _j(s) = T_+ s \text{ for } s>>0 & j = 3 \\
\end{array}\right.}In... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.044527240097522736,
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... |
6403a965c0d6323f05201d3994971d25cb39346d | subsection | 88 | 279 | From Floer cylinders to pseudoholomorphic curves | From this, we obtain the same statement for \tilde{\mathbf {e}} and hence for \mathbf {e}.[Proof of Proposition REF ]
Using the fact that X_H = h^{\prime }(\operatorname{e}^r) R, the Floer equation (REF ) satisfied by \tilde{v} = (b,v) is equivalent to:{\left\lbrace \begin{array}{ll}
db - v {}^{*}\alpha \circ i + h^{\p... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02024451084434986,
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0... |
aabceb8d163d0c48924c1146c183c3ce3476fc7e | subsection | 89 | 279 | From Floer cylinders to pseudoholomorphic curves | Suppose \tilde{v} = (b, v)\colon \mathbb {R}\times S^1 \setminus \Gamma \rightarrow \mathbb {R}\times Y
is the upper level of a split Floer cylinder as in Cases (1) and (2) of Proposition REF .
Then, there exists a pair (\tilde{u}, {\bf e}) consisting of a map \tilde{u} = (a, u)\colon \mathbb {R}\times S^1 \setminus \G... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.009424209594726562,
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... |
3c6f2291dd96f59138941b55efa6ca6a47a37f7a | subsection | 90 | 279 | From Floer cylinders to pseudoholomorphic curves | Then, there exists a pair (\tilde{u}, \bf e) as above, with the following difference:If \tilde{v} converges to a Hamiltonian orbit at +\infty and to a Reeb orbit at -\infty , then u converges to the corresponding Reeb orbit \gamma _+ at +\infty and to the same Reeb orbit \gamma _- at -\infty .The pair (\tilde{u},{\bf e... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04402369260787964,
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-0.003279658267274499,
0.0... |
984d900728b19dc5e591502d5cb8d45d81044186 | subsection | 91 | 279 | From Floer cylinders to pseudoholomorphic curves | In the case of a punctured
cylinder, W^{1,p,\delta }_{(0, }(\mathbb {R}\times S^1 \setminus \lbrace P \rbrace ,
denotes the space of functions exponentially decaying to 0
at -\infty , decaying to a free constant at P and decaying to a free
constant at +\infty .In the case with \Gamma = \lbrace P\rbrace , we fix a para... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.028403345495462418,
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0.002447709208354354,
0.... |
009b479cb51016327c49b84448ab9c5636fac22c | subsection | 92 | 279 | From Floer cylinders to pseudoholomorphic curves | We say \tilde{\mathbf {e}}\in \tilde{\mathcal {X}}^2 if
\tilde{\mathbf {e}}+ (T_+ s + i 0) \, \mu _{+\infty } + (T_- s + i 0) \, \mu _{-\infty } \in W^{1,p,\delta }_{( }( \mathbb {R}\times S^1 \setminus \lbrace P \rbrace ,
Case 3:
\Gamma = \emptyset and the +\infty asymptotic limit of \tilde{v} is a Hamiltonian orbi... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
ce36ff1c90eacb8ae436749bfa81e55e76460586 | subsection | 93 | 279 | From Floer cylinders to pseudoholomorphic curves | We can
write \tilde{\mathcal {C}}^j(\tilde{v}) as the preimage of zero under
the operator\tilde{\mathcal {X}}^j &\rightarrow L^{p,\delta }(T^*( \mathbb {R}\times S^1 \setminus \Gamma )) \\
\tilde{\mathbf {e}}= (e_1,e_2) &\mapsto d e_1 - de_2 \circ i + h^{\prime }(\operatorname{e}^b) ds.Observe that this map descends to... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.014276... |
918db62e69ced7ef57300954523bde0d6a257801 | subsection | 94 | 279 | From Floer cylinders to pseudoholomorphic curves | We need to compute the Conley–Zehnder index of the perturbed asymptotic operator A+\delta , which is -1 by Corollary REF .
Therefore using the fact that all vector spaces in \mathbf {V} are the kernels of the corresponding asymptotic operators, which we identified with , Theorem \ref {T:RiemannRochMB} implies that
\op... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-0.0038748... |
10912489272f8f471770e84b1fbc0e8631fad1f0 | subsection | 95 | 279 | From Floer cylinders to pseudoholomorphic curves | By Lemma REF , it follows that \tilde{\mathbf {e}} is
smooth and satisfies the equationde_1 - d \tilde{e_2} \circ i + h^{\prime }(\operatorname{e}^b) ds =0.For each c >> 1, let S_c := [-c,c] \times S^1 \subset \mathbb {R}\times S^1.
Then,\int _{\partial S_c} \tilde{e}_2 \, dt
&= \int _{S_c} d\tilde{e}_2 \wedge dt = \in... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0109... |
14bad7a8b7dcc348dc8d5dbbc0ab719476880b7a | subsection | 96 | 279 | From Floer cylinders to pseudoholomorphic curves | Let \nu _j: \mathbb {R}\rightarrow \mathbb {R} be smooth functions such that{\left\lbrace \begin{array}{ll}
\nu _j(s) = T_- s \text{ for } s<<0 \text{ and } \nu _j(s) = T_+ s \text{ for } s>>0 & j = 1, 2 \\
\nu _j(s) = 0 \text{ for } s<<0 \text{ and } \nu _j(s) = T_+ s \text{ for } s>>0 & j = 3 \\
\end{array}\right.}In... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.044527240097522736,
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0.006374659016728401,
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... |
c53c5c4ae24f7f59929931dd52ebef2f05b20be8 | subsection | 97 | 279 | From Floer cylinders to pseudoholomorphic curves | From this, we obtain the same statement for \tilde{\mathbf {e}} and hence for \mathbf {e}.[Proof of Proposition REF ]
Using the fact that X_H = h^{\prime }(\operatorname{e}^r) R, the Floer equation (REF ) satisfied by \tilde{v} = (b,v) is equivalent to:{\left\lbrace \begin{array}{ll}
db - v {}^{*}\alpha \circ i + h^{\p... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.026742074638605118,
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0.0015407584141939878,
-0.0000561933484277688,... |
c4a9839a51f06f962ab8a91f1bf0b4ead440ceae | subsection | 98 | 279 | From pseudoholomorphic curves to Floer cylinders | Proposition REF implies that every split Floer cylinder \tilde{v} \colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y has an underlying J_Y-holomorphic curve \tilde{u} \colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y.
We will show that every J_Y-holomorphic curve \ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.008703821338713169,
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0.... |
ead2f9175a4db800db1978535bb158db8e4cf076 | subsection | 99 | 279 | From pseudoholomorphic curves to Floer cylinders | The original J_Y-holomorphic curve is given by
(a, u) = (b - e_1, \phi _R^{-e_2} \circ v).The pair (\tilde{v},{\bf e}) with these properties is unique.In the case that \Gamma = \emptyset and \tilde{u}(s,t)=(Ts + c,\gamma (Tt)) is a trivial cylinder, for some Reeb orbit \gamma of period T, there also exists a pair (\ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
9f546349ab5082fca5533ce070a8117a07312772 | subsection | 100 | 279 | From pseudoholomorphic curves to Floer cylinders | Recall that we have chosen cylindrical coordinates near P, \varphi \colon (-\infty , -1] \times S^1 \rightarrow \mathbb {R}\times S^1 by \varphi (\rho , \theta ) = P +
\operatorname{e}^{2\pi (\rho + i \theta )}, and also a bump function \mu _P that is
identically 1 near P and supported in the image of \varphi .Similarl... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0... |
001e74e253398ad9026bcd520477aeea36026511 | subsection | 101 | 279 | From pseudoholomorphic curves to Floer cylinders | Define the space \mathcal {Y}^3 by the condition that \mathbf {f} admits a lift \tilde{\mathbf {f}} with
\tilde{\mathbf {f}}- (b_+ + i 0) \, \mu _{+\infty } \in W^{1,p, \delta }_{( i\mathbb {R})}( \mathbb {R}\times S^1,
(this corresponds to (2') in Proposition REF ).As before, we notice that the spaces \mathcal {Y}^... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.030157623812556267,
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0.0323... |
78d2a98fb93e11dec0ffac9014475f679371ad89 | subsection | 102 | 279 | From pseudoholomorphic curves to Floer cylinders | Denote the space of solutions to (REF ) in \mathcal {Y}^1 for fixed \tau by \mathcal {C}^{1,\tau }(\tilde{u}).In Case 2, it will be more convenient to study functions \mathbf {g}=(g_1,g_2)=(f_1 - \mu _P \, a, f_2).
We think of {\bf g} as an element of the subspace \hat{ \mathcal {Y}^2}
\subset W^{1,p}_\text{loc}(\mathb... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-0.01783585362136364,
0.0359... |
82e2554e7487e19d57692e5e99585bf03f28aad0 | subsection | 103 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 3, we will study solutions directly and do not require any deformation argument.We need the following smoothness in order to apply the implicit function
theorem, which we will need in the proof of Proposition REF .Lemma 6.13
The nonlinear operators \mathcal {F} and \mathcal {G} are C^1 in
\mathcal {Y}^1 \times... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02688131108880043,
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-0.04430381953716278,
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-0.010389428585767746,
0.03951339051127434,
-0.01... |
47ffcb54f88bf703e05e9587b35268a43cef4ab4 | subsection | 104 | 279 | From pseudoholomorphic curves to Floer cylinders | Hence, there exists a C (depending on R)
so that&0 \le h^{\prime }(\operatorname{e}^{g_1+ \tau \mu _P a}) \le C,\\
&0 \le h^{\prime \prime }(\operatorname{e}^{g_1+ \tau \mu _P a})\operatorname{e}^{g_1+ \tau \mu _P a} \le C\\
&| h^{\prime \prime \prime }(\operatorname{e}^{g_1+ \tau \mu _P a})\operatorname{e}^{2(g_1+ \ta... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.022303495556116104,
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-0.039115019142627716,
0.044790055602788925,
-0.0018... |
15e276cd898634fa5cfa837965e47079b37b88af | subsection | 105 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 1, writing a solution to (REF ) as {\bf f} = f_1 + i f_2, the linearization of the operator \mathcal {F}^\tau is\begin{aligned}D^{1,\tau } \colon T_{\bf f}\mathcal {Y}^1 &\rightarrow L^{p,\delta }(\mathbb {R}\times S^1, (\mathbb {R}^2)^*) \\
F = F_1 + iF_2 &\mapsto dF_1 - dF_2 \circ i + h^{\prime \prime }(\oper... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03619544953107834,
0.002866871887817979,
-0.0011396912159398198,
-0.028260527178645134,
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-0.05163802579045296,
0.006401345133781433,
0.... |
35641428e5d81983baa208e6c07cad7c3d26d5aa | subsection | 106 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 3, the linearized operatorD^3\colon T_{\bf f} \mathcal {Y}^3 \rightarrow L^{p,\delta }(\mathbb {R}\times S^1, (\mathbb {R}^2)^*),where T_{\bf f} \mathcal {Y}^3 = W^{1,p,\delta }_{\mathbf {V}}(\mathbb {R}\times S^1, with V_- = and V+ = iR, is also given by (\ref {E:cylLinear1})
Evaluating (REF ) and (REF ) at \p... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04787791892886162,
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0.03146088868379593,
-0.0093... |
c22e997b6c51818a2f78a9b16a0efd18bf173850 | subsection | 107 | 279 | From pseudoholomorphic curves to Floer cylinders | Then,
the linearized operator D^{1,\tau } is Fredholm of index 0 and is an isomorphism.Fix \tau \in [0,1] and {\bf g} \in \mathcal {C}^{2,\tau }(\tilde{u}). Then, the linearized operator D^{2,\tau } is Fredholm of index 0
and is an isomorphism.For fixed {\bf f} \in \mathcal {C}^{3}(\tilde{u}), D^3 is Fredholm of index
... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.029704617336392403,
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0.023479919880628586,
... |
1f535c5323fef0c8939250e931c6ba65c651a6d2 | subsection | 108 | 279 | From pseudoholomorphic curves to Floer cylinders | We now have V_-=0, V_+=i\mathbb {R} and V_P =, so Theorem \ref {T:RiemannRochMB} implies
\operatorname{Ind}\widehat{D}^{2,\tau } = -1 + (0+1) - (0+1)- (-1 + 0) = 0,
c_1(E,l,\textbf {A}_\Gamma ) = \frac{1}{2}(0-2) = -1 < 0 = \operatorname{Ind}\widehat{D}^{2,\tau }
and we can apply once more \cite {WendlAutomatic}*{P... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04881959781050682,
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0... |
0a7ee9cd552c748f0c35539464c992539d8b0b48 | subsection | 109 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 2, \partial S_c also contains a
small loop around P. In that case, for c sufficiently large, we obtain:}
&= \int _{\partial D_P(1/c)} (f_1- \tau y) \, ds \\
&= \int _{\partial D_P(1/c)} f_1 \, ds.Notice that \int _{\partial D_P(1/c)} f_1 \, ds decays like \frac{1}{c^{2\pi \delta +1}}.The result now follows by t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03893769159913063,
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... |
0409a81e9853d98c2e7a1037b77ed77067dc99d5 | subsection | 110 | 279 | From pseudoholomorphic curves to Floer cylinders | These and the asymptotic behavior of \nu imply that there are constants C and \tilde{C} such that:F(x,s)\le C for all (x,s);
F(x,s) \ge \tilde{C} for x\le \ln 2, uniformly in s.The claim now follows from standard existence and uniqueness theory for solutions to ODE's.The facts that for s<<0 we have that b_- is the onl... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0... |
3046d824f64b5d2aad075d2b430a00e1e955b90d | subsection | 111 | 279 | From pseudoholomorphic curves to Floer cylinders | The projection of these curves to the first cylinder factor is the identity.Since \lim _{s\rightarrow -\infty } f_2^1 = \lim _{s\rightarrow -\infty } f_2^0, Lemma REF implies that they have lifts \tilde{f}_2^1 and \tilde{f}_2^0 to \mathbb {R}, with the same limits as s\rightarrow \pm \infty .
By assumption, {\bf f}^1 a... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.08826813101768494,
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-... |
ea374ca7e603a4098008b488256292f2bebabb8f | subsection | 112 | 279 | From pseudoholomorphic curves to Floer cylinders | Then,({\bf g} \circ \varphi _P)(\rho ,\theta ) = (g_1(\operatorname{e}^{2\pi \rho } \cos (2\pi \theta ) + P_s),0)and for \rho << -1 the Mean Value Theorem gives|g_1(\operatorname{e}^{2\pi \rho } \cos (2\pi \theta ) + P_s) - g_1(P_s)| \le ||g_1^{\prime }||_{L^\infty }\,\operatorname{e}^{2\pi \rho }where the sup norm is ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.053719308227300644,
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0.... |
4ffd2a9cef4215524911facaaff09af8a0b6c762 | subsection | 113 | 279 | From pseudoholomorphic curves to Floer cylinders | Lemma REF guarantees the existence and uniqueness of solutions to (REF ) in \mathcal {Y}^1 (which corresponds to \tau = 0).
Therefore, we conclude that (REF ) (which corresponds to \tau = 1) also has a unique solution in \mathcal {Y}^1, which finishes the proof.Case 2 works the same way, with solution \bf g to (REF ) i... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03908681124448776,
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0.... |
349f862fc34118f1af102f22cd82f5da817ccd54 | subsection | 114 | 279 | From pseudoholomorphic curves to Floer cylinders | In particular then, we again obtain the claimed equality.Proposition REF implies that every split Floer cylinder \tilde{v} \colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y has an underlying J_Y-holomorphic curve \tilde{u} \colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.042584121227264404,
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0... |
9a783c4284a0f8ec018a61a6773e644cbddc0613 | subsection | 115 | 279 | From pseudoholomorphic curves to Floer cylinders | The original J_Y-holomorphic curve is given by
(a, u) = (b - e_1, \phi _R^{-e_2} \circ v).The pair (\tilde{v},{\bf e}) with these properties is unique.In the case that \Gamma = \emptyset and \tilde{u}(s,t)=(Ts + c,\gamma (Tt)) is a trivial cylinder, for some Reeb orbit \gamma of period T, there also exists a pair (\ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05700834095478058,
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... |
0265c648c770d4bb53742d3c3966d0372fb126f3 | subsection | 116 | 279 | From pseudoholomorphic curves to Floer cylinders | Recall that we have chosen cylindrical coordinates near P, \varphi \colon (-\infty , -1] \times S^1 \rightarrow \mathbb {R}\times S^1 by \varphi (\rho , \theta ) = P +
\operatorname{e}^{2\pi (\rho + i \theta )}, and also a bump function \mu _P that is
identically 1 near P and supported in the image of \varphi .Similarl... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.037026409059762955,
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0.01278983149677515,
0.0035027044359594584,
0.0... |
ca9182a6887231dc9d1b80e0b7346149aa4aac63 | subsection | 117 | 279 | From pseudoholomorphic curves to Floer cylinders | Define the space \mathcal {Y}^3 by the condition that \mathbf {f} admits a lift \tilde{\mathbf {f}} with
\tilde{\mathbf {f}}- (b_+ + i 0) \, \mu _{+\infty } \in W^{1,p, \delta }_{( i\mathbb {R})}( \mathbb {R}\times S^1,
(this corresponds to (2') in Proposition REF ).As before, we notice that the spaces \mathcal {Y}^... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02149321883916855,
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0.030157623812556267,
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0.0323... |
e0b7053fd08660e5c8b8180397fcafbdbfb8db3b | subsection | 118 | 279 | From pseudoholomorphic curves to Floer cylinders | Denote the space of solutions to (REF ) in \mathcal {Y}^1 for fixed \tau by \mathcal {C}^{1,\tau }(\tilde{u}).In Case 2, it will be more convenient to study functions \mathbf {g}=(g_1,g_2)=(f_1 - \mu _P \, a, f_2).
We think of {\bf g} as an element of the subspace \hat{ \mathcal {Y}^2}
\subset W^{1,p}_\text{loc}(\mathb... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.046107739210128784,
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0.0359... |
bc23dff03b9c7284acf017cd4a8f4e41cc00ffa2 | subsection | 119 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 3, we will study solutions directly and do not require any deformation argument.We need the following smoothness in order to apply the implicit function
theorem, which we will need in the proof of Proposition REF .Lemma 6.13
The nonlinear operators \mathcal {F} and \mathcal {G} are C^1 in
\mathcal {Y}^1 \times... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02688131108880043,
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0.03951339051127434,
-0.01... |
36d05af2f35a6e7db9b6a2da94b24a69e514070f | subsection | 120 | 279 | From pseudoholomorphic curves to Floer cylinders | Hence, there exists a C (depending on R)
so that&0 \le h^{\prime }(\operatorname{e}^{g_1+ \tau \mu _P a}) \le C,\\
&0 \le h^{\prime \prime }(\operatorname{e}^{g_1+ \tau \mu _P a})\operatorname{e}^{g_1+ \tau \mu _P a} \le C\\
&| h^{\prime \prime \prime }(\operatorname{e}^{g_1+ \tau \mu _P a})\operatorname{e}^{2(g_1+ \ta... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.022303495556116104,
0.05610961094498634,
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0.044790055602788925,
-0.0018... |
49ab44ba6e3f77c68c8f0c54644acc7da7f0853d | subsection | 121 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 1, writing a solution to (REF ) as {\bf f} = f_1 + i f_2, the linearization of the operator \mathcal {F}^\tau is\begin{aligned}D^{1,\tau } \colon T_{\bf f}\mathcal {Y}^1 &\rightarrow L^{p,\delta }(\mathbb {R}\times S^1, (\mathbb {R}^2)^*) \\
F = F_1 + iF_2 &\mapsto dF_1 - dF_2 \circ i + h^{\prime \prime }(\oper... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03619544953107834,
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0.006401345133781433,
0.... |
cacd0ec363b6b116f0c21c6cf53444a1624cc43a | subsection | 122 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 3, the linearized operatorD^3\colon T_{\bf f} \mathcal {Y}^3 \rightarrow L^{p,\delta }(\mathbb {R}\times S^1, (\mathbb {R}^2)^*),where T_{\bf f} \mathcal {Y}^3 = W^{1,p,\delta }_{\mathbf {V}}(\mathbb {R}\times S^1, with V_- = and V+ = iR, is also given by (\ref {E:cylLinear1})
Evaluating (REF ) and (REF ) at \p... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04787791892886162,
0.011504127644002438,
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0.03146088868379593,
-0.0093... |
b4d4f2901682118f35c1c12e95cd30198b1cef80 | subsection | 123 | 279 | From pseudoholomorphic curves to Floer cylinders | Then,
the linearized operator D^{1,\tau } is Fredholm of index 0 and is an isomorphism.Fix \tau \in [0,1] and {\bf g} \in \mathcal {C}^{2,\tau }(\tilde{u}). Then, the linearized operator D^{2,\tau } is Fredholm of index 0
and is an isomorphism.For fixed {\bf f} \in \mathcal {C}^{3}(\tilde{u}), D^3 is Fredholm of index
... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.029704617336392403,
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0.034144289791584015,
0.03747022897005081,
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-0.018262160941958427,
0.023479919880628586,
... |
b2b6074dddf4b8fca66d08991ff7f91cbe81b528 | subsection | 124 | 279 | From pseudoholomorphic curves to Floer cylinders | We now have V_-=0, V_+=i\mathbb {R} and V_P =, so Theorem \ref {T:RiemannRochMB} implies
\operatorname{Ind}\widehat{D}^{2,\tau } = -1 + (0+1) - (0+1)- (-1 + 0) = 0,
c_1(E,l,\textbf {A}_\Gamma ) = \frac{1}{2}(0-2) = -1 < 0 = \operatorname{Ind}\widehat{D}^{2,\tau }
and we can apply once more \cite {WendlAutomatic}*{P... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04881959781050682,
0.03200734779238701,
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0.04192383214831352,
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-0.01845991052687168,
-0.008322215639054775,
0... |
0cc95f7e497fdd757b47bd48fb29b7284cf50500 | subsection | 125 | 279 | From pseudoholomorphic curves to Floer cylinders | In Case 2, \partial S_c also contains a
small loop around P. In that case, for c sufficiently large, we obtain:}
&= \int _{\partial D_P(1/c)} (f_1- \tau y) \, ds \\
&= \int _{\partial D_P(1/c)} f_1 \, ds.Notice that \int _{\partial D_P(1/c)} f_1 \, ds decays like \frac{1}{c^{2\pi \delta +1}}.The result now follows by t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03893769159913063,
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0.02964574284851551,
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-0.023908840492367744,
-0.0029161309357732534,
... |
9c3c84e3504599a58ac74d4865d25bb05d35cb23 | subsection | 126 | 279 | From pseudoholomorphic curves to Floer cylinders | These and the asymptotic behavior of \nu imply that there are constants C and \tilde{C} such that:F(x,s)\le C for all (x,s);
F(x,s) \ge \tilde{C} for x\le \ln 2, uniformly in s.The claim now follows from standard existence and uniqueness theory for solutions to ODE's.The facts that for s<<0 we have that b_- is the onl... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.048539966344833374,
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0.023232674226164818,
0.0... |
689ac0a301c0dae4fa98d6d02515c63d158af2f6 | subsection | 127 | 279 | From pseudoholomorphic curves to Floer cylinders | The projection of these curves to the first cylinder factor is the identity.Since \lim _{s\rightarrow -\infty } f_2^1 = \lim _{s\rightarrow -\infty } f_2^0, Lemma REF implies that they have lifts \tilde{f}_2^1 and \tilde{f}_2^0 to \mathbb {R}, with the same limits as s\rightarrow \pm \infty .
By assumption, {\bf f}^1 a... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.08826813101768494,
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-... |
9651810585838426dfbfa0bc0608a5ae3e6bef3a | subsection | 128 | 279 | From pseudoholomorphic curves to Floer cylinders | Then,({\bf g} \circ \varphi _P)(\rho ,\theta ) = (g_1(\operatorname{e}^{2\pi \rho } \cos (2\pi \theta ) + P_s),0)and for \rho << -1 the Mean Value Theorem gives|g_1(\operatorname{e}^{2\pi \rho } \cos (2\pi \theta ) + P_s) - g_1(P_s)| \le ||g_1^{\prime }||_{L^\infty }\,\operatorname{e}^{2\pi \rho }where the sup norm is ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.... |
463f0214fbafc14b628c4d54abab7f18611ce376 | subsection | 129 | 279 | From pseudoholomorphic curves to Floer cylinders | Lemma REF guarantees the existence and uniqueness of solutions to (REF ) in \mathcal {Y}^1 (which corresponds to \tau = 0).
Therefore, we conclude that (REF ) (which corresponds to \tau = 1) also has a unique solution in \mathcal {Y}^1, which finishes the proof.Case 2 works the same way, with solution \bf g to (REF ) i... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03864011913537979,
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0.032444268465042114,
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0.02531... |
597a161ef361ca7a9cfa121fa09d4af6083fc8c2 | subsection | 130 | 279 | Pseudoholomorphic spheres in | The previous sections explain that split Floer cylinders are in
some sense equivalent to J_Y-holomorphic curves.
We will now describe J_Y-holomorphic curves in a manner that is more suitable
for computing the Floer differential.Let \tilde{u} = (a,u): \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03666205704212189,
0.03980626165866852,
-0.01776629313826561,
0.041027314960956573,
0.034372586756944656,
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0.05467255786061287,
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0.00614341301843524,
0.01598050445318222,
0.01866... |
9bb432f5d92c4cc8b637cda2e2f714dfb7b5a4b1 | subsection | 131 | 279 | Pseudoholomorphic spheres in | This proves the following result.Lemma 7.1
Every J_Y-holomorphic curve \tilde{u} = (a,u): \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y defines a J_\Sigma -holomorphic map w: \mathbb {CP}^1 \rightarrow \Sigma and a meromorphic section of w^*E \rightarrow \mathbb {CP}^1, with zero at 0 (and at P... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05666022002696991,
0.03490538150072098,
-0.007605040445923805,
0.007616482209414244,
0.027826666831970215,
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0.014378637075424194,
... |
3613836e416742cde4a51a47e84b3966efb525ff | subsection | 132 | 279 | Pseudoholomorphic spheres in | Then, the action of the torus of rotation in the
domain and in the fibre can be
seen as a linear map 2 \rightarrow 2
represented by the matrix \begin{pmatrix} k_- &
1\\ k_+ & 1 \end{pmatrix}. See Equation (REF ) for the analogous discussion
in the case of lifts to solutions of Floer's equation in \mathbb {R}\times Y, i... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.022345218807458878,
0.007726958487182856,
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0.038127291947603226,
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0.006834065541625023,
-0.012744711712002754,
0.... |
6180c2c0ff6ca7a25f44ffb05ab491aa07cfbb2d | subsection | 133 | 279 | Pseudoholomorphic spheres in | Since punctures of finite energy pseudoholomorphic curves in \mathbb {R}\times Y are asymptotic to Reeb orbits in Y, and these are
multiple covers of the fibres of Y\rightarrow \Sigma , w extends to a J_\Sigma -holomorphic map \mathbb {CP}^1 \rightarrow \Sigma
*Theorem 4.1.2. We use the standard identification of \mat... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05123084783554077,
0.02869720757007599,
0.014760525897145271,
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0.040795501321554184,
0.0... |
c067410d257aad94aa357ca07dcf2076e71bd4a2 | subsection | 134 | 279 | Pseudoholomorphic spheres in | The count of maps w is
related with the computation of genus zero Gromov–Witten numbers of \Sigma . More about this will be said below.
We can also give a complete description of the relevant meromorphic sections s. Given a divisor of points D in \mathbb {CP}^1 and a holomorphic line bundle L over \mathbb {CP}^1, where... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07072607427835464,
0.020824220031499863,
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0.02970311976969242,
0.011853787116706371,
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0.04146537184715271,
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0.018474821001291275,
0.011640205048024654,
0.... |
3f7d6a0710e244de8988712e75e3ad998e244e7e | subsection | 135 | 279 | Pseudoholomorphic spheres in | See Equation (REF ) for the analogous discussion
in the case of lifts to solutions of Floer's equation in \mathbb {R}\times Y, instead of pseudoholomorphic curves.The number of lifts we want is the absolute
value of the degree of this map T^2 \rightarrow T^2,
which is the absolute value of the determinant
of the matrix... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02768898569047451,
-0.008062605746090412,
-0.02334112860262394,
0.034294676035642624,
0.027841541916131973,
0.003192242467775941,
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0.02369200810790062,
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0.048268843442201614,
-0.01676594838500023,
0.015476846136152744,
-0.02039678953588009,
0.03... |
c26edaaa59a8322ab69ab13a215ea2a0d5f69edd | subsection | 136 | 279 | Pseudoholomorphic spheres in | We use the standard identification of \mathbb {R}\times S^1 with * \subset \mathbb {CP}^1.The symplectization \mathbb {R}\times Y can be identified with the complement of the zero section in a complex line bundle E \rightarrow \Sigma (the dual to the normal bundle to \Sigma in X).
We have the following commutative diag... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05223271995782852,
0.018031271174550056,
-0.004545954987406731,
0.03188270702958107,
0.013485316187143326,
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0.033347174525260925,
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-0.008550361730158329,
0.03545234724879265,
0.0... |
846abf33c36b17b372924a4f1a86fb9843532794 | subsection | 137 | 279 | Pseudoholomorphic spheres in | Given a divisor of points D in \mathbb {CP}^1 and a holomorphic line bundle L over \mathbb {CP}^1, where both D and L have degree d, there is a *-family of meromorphic sections of L, such that the divisor associated with each section is D.
One can justify this fact by reducing it to the simplest case of trivial L: use ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04018932208418846,
0.022825825959444046,
-0.026045244187116623,
0.047848790884017944,
0.032346758991479874,
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0.04476669430732727,
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0.00490923086181283,
0.01021516788750887,
0.... |
1c56b248268af66ac8d5a021ad374393fec8e175 | subsection | 138 | 279 | Pseudoholomorphic spheres in | The difference between \tilde{v} and \tilde{u} is given by {\bf e} \colon \mathbb {R}\times S^1 \setminus \Gamma \rightarrow \mathbb {R}\times S^1 solving (REF ), and the information lost when we project \tilde{u} to w is a meromorphic section of a line bundle L over \mathbb {CP}^1. Alternatively, we could have project... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05731365829706192,
0.009178425185382366,
-0.005748914089053869,
0.020737290382385254,
0.02864157035946846,
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0.043397244065999985,
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-0.0035515851341187954,
-0.030793121084570885,
0... |
20af3715ebedbac647bbc93ca1a71c4de2b4e281 | subsection | 139 | 279 | Orientations | We want to determine the signs of the contributions to the split symplectic homology differential. The orientation conventions
for Cases 0 through 3 are explained in *Section 7. As we saw before, Cases 1 through 3 can be related to counts
of pseudoholomorphic spheres in \Sigma and X.
We wish to compare the signs that a... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.036610912531614304,
-0.0426517128944397,
-0.01341637410223484,
0.003767872927710414,
0.007528118789196014,
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0.004099659156054258,
0.014423173852264881,
0.0003939963353332132,
0.020669911056756973,
-0.04521447420120239,
0.018427493050694466,
0.010998528450727463,
-0... |
d595b3af8bd92021103c2054d05b51933ef8de4b | subsection | 140 | 279 | Orientations | A simple calculation yields the following.Claim Let s_{12} \colon (V_1 \oplus V_2)/\ker f \rightarrow V_1 \oplus V_2 be a right-inverse for the projection \pi _{12} \colon V_1 \oplus V_2 \rightarrow (V_1 \oplus V_2)/\ker f. Then,
s_{21}r \circ s_{12} \circ r^{-1} \colon (V_2 \oplus V_1)/\ker g \rightarrow V_2 \oplus V_... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07228308916091919,
0.006124068517237902,
-0.016666624695062637,
0.0004619283718056977,
0.018070772290229797,
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0.02786928229033947,
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-0.01784183457493782,
0.001265831640921533,
0.... |
37cb388f23f511ae27232796de5921be29bebc6f | subsection | 141 | 279 | Orientations | This is the reason for the name used for this orientation convention, and for the sign in Part (1) above.Lemma 8.5
The identification of \ker f with the fibre sum orientation (as in Definition REF ) and (f_1,f_2)^{-1}(\Delta ) with preimage orientation (as in Definition REF ) changes orientations by (-1)^{(\dim V_1 + ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03527453541755676,
-0.00576338917016983,
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-0.009314491413533688,
-0.0074264188297092915,
0.032863907516002655,
0.023099327459931374,
0.0293395034968853,
0.011519149877130985,
0.017423667013645172,
-0.05324745923280716,
0.007147975731641054,
0.006110489368438721,
0.... |
eed1aa1c9f9023be5c0e1b3d1c8d0ba9bb4320b7 | subsection | 142 | 279 | Orientations | \\
&\hspace{142.26378pt} .(f_1,f_2)(H) \oplus \Delta \oplus (f_1,f_2)^{-1}(\Delta ) \cong \\
&\stackrel{(P2)}{\cong } (-1)^{\dim V_1 \dim W + \dim V_1 \dim V_2 + \dim V_2 \dim W} W \oplus W \oplus (f_1,f_2)^{-1}(\Delta ) \cong \\
&\stackrel{\varphi }{\cong } (-1)^{\dim V_1 \dim W + \dim V_1 \dim V_2 + \dim V_2 \dim W +... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.043365661054849625,
-0.011291551403701305,
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0.0026149859186261892,
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0.02325449138879776,
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0.011047408916056156,
0.0005035421345382929,
0... |
8675dfb9b172292c6e1beb65fdd5035cac67e91b | subsection | 143 | 279 | Orientations | In *Section 7, these spaces are oriented using the fibre sum rule.
the spaces of cylinders are given a coherent orientation and the critical manifolds are oriented as follows.For any oriented manifold S with a Morse–Smale pair (f_S,Z_S), fixing an orientation on a critical manifold of a critical point p
also fixes an o... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.00745584350079298,
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... |
24dc97eafd7d9b4207b32c07efde58ccc2e8b732 | subsection | 144 | 279 | Orientations | Under our conventions, these maps
are always orientation-preserving for W^u_Y(\widehat{p}), and preserve orientations for W^s_Y(p) iff M(p) is even.Each cascade in (REF ) projects to a chain of pearls in \Sigma from q to p, with exactly one sphere, contained in\mathcal {M}(q,p;A)W^s_{\Sigma }(q) \times _{\operatorname{... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-... |
c282be73d5d2b23800885553ba7e8e020b85dddf | subsection | 145 | 279 | Orientations | To compare the two signs, it will be useful to also consider the fibre product\overline{\mathcal {M}({q}_{k_-},\widehat{p}_{k_+};A)}\pi _\Sigma ^{-1}\left(W^s_{\Sigma }(q) \right) \times _{\operatorname{ev}} \mathcal {M}^*_{H,0,\mathbb {R}\times Y;k_-,k_+}(A;J_Y) \times _{\operatorname{ev}} \pi _\Sigma ^{-1}\left(W^u_{... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.001875... |
ecfa7a333f83f90f3fe997600133191ba0b3dc12 | subsection | 146 | 279 | Orientations | Recall that the left sides in (REF ) were given orientations in (REF ), which agree with
the fibre sum orientations on the right sides.Note that \mathcal {M}(\widehat{q}_{k_-},{p}_{k_+};A) is a codimension 2 submanifold of \overline{\mathcal {M}({q}_{k_-},\widehat{p}_{k_+};A)}.
Given an element (x,u,y) \in \mathcal {M}... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.... |
47314c98b4d82c2e64d089270404f4beb0b6b45e | subsection | 147 | 279 | Orientations | \\
&\qquad .\lbrace 1\rbrace {}{i\,}\times {}{{s}}{} \big ( \pi _\Sigma ^{-1}\left(W^s_{\Sigma }(q) \right) \times _{\operatorname{ev}} \mathcal {M}^*_{H;k_-,k_+}(A) \times _{\operatorname{ev}} \pi _\Sigma ^{-1}\left(W^u_{\Sigma }(p) \right) \big ) {}{\widehat{u}\,}\times {}{i} \lbrace 1\rbrace = \\
&= (-1)^{M(q)} \lbr... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.025652039796113968,
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0.... |
3218e468f738946e50ffe72e46947d4d95c05745 | subsection | 148 | 279 | Orientations | The last identity follows from the fact that the fibre sum
orientation on the zero dimensional vector space
0 {}{di\,}\oplus {}{d{s}}{} \, \mathbb {R}^2 {}{d\widehat{u}\,}\oplus {}{di} \,0
is negative, which is a simple calculation (the reason ultimately being the fact that the matrix in (REF ) has
negative determinant... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0038... |
5a3c239f9ef210cfbd938bf9152616c1ab0a5de6 | subsection | 149 | 279 | Orientations | This can be identified with T_{(a,b)} (A {}{f_1\,}\times {}{f_2}B), via the restriction of (\pi _A,\pi _B) to the fibre sum.
This gives an identificationT_{(\tilde{a}, \tilde{b})} (\tilde{A} {}{\tilde{f}_1\,}\times {}{\tilde{f}_2}{} \tilde{B}) \cong \left( T_{(a,b)} (A {}{f_1\,}\times {}{f_2}B) \right) \oplus \mathbb {... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.010773732326924801,
0.013772369362413883,
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0.027773642912507057,
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0.03528168424963951,
0.0... |
6e8dbeeafd11acd9fdf05b08f76911b8b8bca647 | subsection | 150 | 279 | Orientations | The commutative diagram
T AT B(d f) R [df]id T RTATB(d f) [df] T Y
defines the isomorphism \varphi .Denote by \psi the composition T A \oplus T B \oplus \mathbb {R}\oplus \mathbb {R}\rightarrow T A \oplus \mathbb {R}\oplus T B \oplus \mathbb {R}\rightarrow T \tilde{A} \oplus T\tilde{B}, where the first isomorphism... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.032014429569244385,
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0.022233089432120323,
0.0... |
36493692b5d0847f2444b69af2cd074d3450c657 | subsection | 151 | 279 | Orientations | By Part (2) in Definition REF and the commutativity of the diagram, we conclude that the isomorphism \chi changes orientations by (-1)^{\dim B + 1 + \dim B + 1} = 1, as wanted.The space (REF ) is an S^1-bundle over (REF ). With respect to the connections discussed above (see (REF )), we have a decompositionT_{(x,u,y)}\... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03863558918237686,
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... |
771b891001bf97caba6e82bad44c43e7e3efc008 | subsection | 152 | 279 | Orientations | Writing (REF ) as(W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma )) \times _{\operatorname{ev}} W_{\Sigma }^u(p)we apply Lemma REF to W^s_{\Sigma }(q) \hookrightarrow \Sigma and \operatorname{ev}_\Sigma ^1\colon \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma )\rightarrow \Sigma to get a sign(... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.038617804646492004,
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0.01... |
0555862d09c8166313b546632097ad669effde54 | subsection | 153 | 279 | Orientations | We then apply Lemma REF to \operatorname{ev}^2\colon W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma ) \rightarrow \Sigma and W_{\Sigma }^u(p) \hookrightarrow \Sigma to get the sign(-1&)^{\dim (W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma ) + \dim... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03560476377606392,
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... |
631ee225372749736a4d65ae2340462a518242c6 | subsection | 154 | 279 | Orientations | This oriented zero-dimensional vector space coincides with the quotient of
\left( T_{(x,u,y)}\mathcal {M}(\widehat{q}_{k_-},{p}_{k_+};A) \right) \oplus \mathbb {R}_{\operatorname{domain}} by the infinitesimal
*-action on the domain of u.
On the other hand, by Definition REF ,
the Gromov–witten sign of (\pi _\Sigma (x),... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04355968162417412,
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... |
649d3ee61312e6f3f63ea8c6215f8922fb248b47 | subsection | 155 | 279 | Orientations | The corresponding sign associated to an element in the zero-dimensional
\operatorname{ev}^{-1}(pt) is its Gromov–Witten sign.Observe that this is oriented diffeomeorphic to \mathcal {M}_X^*(B;J_W) \times _{\operatorname{ev}} \lbrace pt\rbrace ,
by Lemma REF . Analogously to what was pointed out in Remark REF ,
this ori... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04145541042089462,
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-0.009432785212993622,... |
737411b77767b31c0f38cc31730bcd9bb140c92a | subsection | 156 | 279 | Orientations | We have oriented isomorphisms\big [&W^s_{Y}(\widehat{p}) \times _{\operatorname{ev}} \big ( \mathcal {M}_X^*(B;J_W) \times _{\tilde{\operatorname{ev}}} \mathcal {M}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y) \big ) \big ]\times _{\operatorname{ev}} W_{Y}^u({p}) \\
& \cong \big [\big (\mathcal {M}_X^*(B;J_W) \times _{\t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0007353468099609017,
0.0312... |
ea57002faf86e9c407027c1ac872aafbdf74e4aa | subsection | 157 | 279 | Orientations | The first isomorphism preserves orientations, by Lemma REF (\dim \big ( \mathcal {M}_X^*(B;J_W) \times _{\tilde{\operatorname{ev}}} \mathcal {M}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y) \big ) + \dim Y is even).
The second and third isomorphisms are orientation-preserving, by the associativity of the fibre sum orient... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.041782476007938385,
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... |
fe3da263639d7fda379b8c90147d4ad93bb5d3c1 | subsection | 158 | 279 | Orientations | Denote by \operatorname{ev}\colon \mathcal {M}^*_{H}(B;J_W)/* \rightarrow X\times \Sigma the evaluation map at (0,\infty )
(where we quotiented by the space * of domain automorphisms). Write W^u_{X}(x) for \psi \big (W^u_{W}(x)\big ).Definition 8.18 The preimage orientation on \operatorname{ev}^{-1}\left(W^u_{X}(x) \ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0712776705622673,
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0.014440135098993778,
0... |
767ca86bc5fdf667a32cf4286c0d0d673d53edb0 | subsection | 159 | 279 | Orientations | The corresponding sign associated to an element in this
zero-dimensional manifold is its Gromov–Witten sign.If W^u_{X}(x) \subset X and W_{\Sigma }^u(p)\subset \Sigma represent homology classes, then the Gromov–Witten orientation determines the signs with which elements in \operatorname{ev}^{-1}\left(W^u_{X}(x) \times ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03924427181482315,
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0.0... |
cb0e15b8b8f4b23bdfe4cad4204b0f4e5249e7c4 | subsection | 160 | 279 | Orientations | We used
the associativity of the fibre sum orientation, the oriented diffeomorphism \mathcal {M}^*_{H,k_+}(0;J_Y)\cong \mathbb {R}_2 \times Y
(see *Lemma 5.22), the fact that W and \mathcal {M}^*_{H}(B;J_W) are
even dimensional and *Proposition 7.5.(a)&(c).
Now, if we think of \mathcal {M}^*_{H}(B;J_W)
as a space of ps... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01922532543540001,
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... |
fc37d902a183db052f7955a03f5f53d2c282a603 | subsection | 161 | 279 | Orientations | Each of the
Cases 1 through 3 motivates one of the following definitions.Definition 8.20
Let q,p\in \operatorname{Crit}(f_\Sigma ) and A \in H_2(\Sigma ;\mathbb {Z}). Definen_A(q,p) := \# \mathcal {M}_A(q,p),where (using the notation of (REF ))\mathcal {M}_A(q,p) := \mathcal {M}(q,p;A) / *.Here, \mathcal {M}(q,p;A) is... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.022386575117707253,
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0.03763142228126526,
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-0.018129006028175354,
0.011338259093463421,
0.0025274551007896... |
fdbc56782bc3595be998189f97cb82850b1abb3f | subsection | 162 | 279 | Orientations | Similarly, the coefficients n_B can
only be non-trivial if\, \langle c_1(TX),B\rangle - B\bullet \Sigma = (1 - K/\tau _X) \, \langle c_1(TX),B\rangle = (\tau _X - K) \, \omega (B) = 1.The coefficients n_B(x,p) can only be non-zero if\, \langle c_1(TX),B\rangle - B\bullet \Sigma = (1 - K/\tau _X) \, \langle c_1(TX),B\ra... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.01491335965692997,
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-0.03356459364295006,
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df1b90ffb9256aea82c029323b89e7369d8053d9 | subsection | 163 | 279 | Orientations | Denoting also by f and g the appropriate isomorphisms in Part (1) of Definition REF , we have an isomorphismg \circ r \circ f^{-1} \colon W \rightarrow W,which is -\operatorname{Id}, hence it changes orientation by (-1)^{\dim W}.Denote also by r the induced map (V_1 \oplus V_2)/\ker f \rightarrow (V_2 \oplus V_1)/\ker ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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